Feb. 7, 2022
Speaker: Yael Davidov
Title: Admissibility Problems for Finite Groups
Abstract: An interesting open problem asks whether every finite group appears as the Galois group of an extension of the rational numbers, this has been called “The inverse Galois problem”. We won’t really be talking about that problem, rather we will try to answer a similar question about of the admissibility of a finite group over a given field. You don’t need to know what admissibility means but as you might expect based on my area of interest, it has to do with division algebras. The talk should be fairly accessible and we will mostly talk about admissibility over the rationals. I will then state some more general results related to this question at the end of the talk.
Feb. 21, 2022
Speaker: Terence Coelho
Title: Finite implications from level 1 affine Lie algebra modules
Abstract: The VOA realization of level 1 affine modules gives a bijection between cosets of the root lattice in the weight lattice and the level 1 modules. In this talk, we’ll get a rough understanding of the structure of these modules and see where this connection comes from, exposing some interesting results about finite root systems you may not have heard of before.
Feb. 28, 2022
Speaker: Nicholas Bakes
Title: Representations of the General Linear Group of a Finite Field
Abstract: For a finite field k, the irreducible complex representations of GL_2(k) can be completely classified. Most of them can be found by taking characters of some subgroups and inducing these characters to GL_2(k). However, some of them, called cuspidal representations, are not components of induced characters. This theory is great practice with the representation theory of finite groups, induced representations, and properties of GL_2.
Mar. 21, 2022
Speaker: Jishen Du
Title: A brief intro to BGG category and BGG resolution
Abstract: The category U(g)-mod for a semisimple Lie algebra is too "big" to be understood categorically/homologically. So we really want to define a full subcategory of U(g)-mod where lots of good categorical properties hold, e.g. finite dimensional/Noetherian/Artinian properties. The first naive attempt is the category of finite dimensional modules. However, it is too trivial categorically speaking. A more complicated one is the BGG category. We are motivated by natural questions: What's Hom(M(lambda),M(mu))? What are the composition factors of Verma M(lambda)? Finally, we will talk about the BGG resolution and some applications of it.
Apr. 11, 2022
Speaker: Brian Pinsky
Title: Golod-Shafarevich groups and algebras
Abstract: Golod-Shaferevich groups are finitely generated infinite torsion groups. You may remember from previous garts talks how to construct such an object. This construction is different ...mumble mumble... taylor series ...mumble mumble... epsilon ...mumble. I've read about these before but forgotten most of it. To prevent myself spending too long on this, I'm only allowing myself 2 hours to prepare this talk. There's a good chance it means I don't have a full hour's worth of stuff to say, but we haven't had a GARTS talk in a few weeks and I miss it.
Apr. 25, 2022
Speaker: Tamar Lichter Blanks
Title: Quadratic forms and algebraic structures
Abstract: A Witt invariant is a rule that assigns quadratic forms to algebraic objects, satisfying some conditions. There is a ring of Witt invariants, called Inv_k(G,W), for any finite group G and field k. These rings are not well understood except in some specific cases. In this talk, we'll define the ring Inv_k(G,W), give some examples of what is and isn't known, and discuss some of the tools one can use to think about Witt invariants, including ideas from representation theory, category theory, and Galois theory.
May. 2, 2022
Speaker: Yifeng Huang
Title: Point count of the variety of modules over the quantum plane over a finite field
Abstract: In 1960, Feit and Fine gave a formula for the number of pairs of commuting n by n matrices over a finite field. We consider a quantum deformation of the problem, namely, counting pairs (A,B) of n by n matrices over a finite field that satisfy AB=qBA for a fixed nonzero scalar q. We give a formula for this count in terms of the order of q as a root of unity, generalizing Feit and Fine's result. In this talk, after explaining the title and the results, we will discuss a curious phenomenon that one sees when comparing the commutative case (q=1) and the general case from a geometric viewpoint.
Spring 2021 Talks (Jason Saied)